𝑆𝐿(2,𝐶) actions on compact Kaehler manifolds

Carrell, James B.; Sommese, Andrew John

doi:10.1090/S0002-9947-1983-0684500-X

$\textrm {SL}(2, \textbf {C})$ actions on compact Kaehler manifolds
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by James B. Carrell and Andrew John Sommese PDF

Trans. Amer. Math. Soc. 276 (1983), 165-179 Request permission

Abstract:

Whenever $G = SL(2,C)$ acts holomorphically on a compact Kaehler manifold $X$, the maximal torus $T$ of $G$ has fixed points. Consequently, $X$ has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the $G$-action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in $G$ preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres $G$-equivariantly over a component of ${X^T}$. We give some applications; e.g. we classify all compact Kaehler manifolds $X$ admitting a $G$-action with no three dimensional orbits. In particular we show that if $X$ is projective and has no three dimensional orbit, and if $\text {Pic}(X) \cong {\mathbf {Z}}$, then $X = C{{\mathbf {P}}^n}$. We also show that if $X$ admits a holomorphic vector field with unirational zero set, and if $\operatorname {Aut}_0(X)$ is reductive, then $X$ is unirational.

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